Solve the system by Gauss - Jordan elimination.
{ 1/3x+ 3/4y− 2/3z=−8 x+ 1/2y+ 1/3z=18 1/6x− 1/8y−z=−24
a. (−6, 8, −24)
b. (−6, −8, −24)
c. (0, 8, −24)
d. (6,8,24)
e. (6, −8, 24)
Solve the following system:
{x/3+(3 y)/4-(2 z)/3 = -8
x+y/2+z/3 = 18
x/6-y/8-z = -24
Express the system in matrix form:
(1/3 | 3/4 | -2/3
1 | 1/2 | 1/3
1/6 | -1/8 | -1)(x
y
z) = (-8
18
-24)
Write the system in augmented matrix form and use Gaussian elimination:
(1/3 | 3/4 | -2/3 | -8
1 | 1/2 | 1/3 | 18
1/6 | -1/8 | -1 | -24)
Swap row 1 with row 2:
(1 | 1/2 | 1/3 | 18
1/3 | 3/4 | -2/3 | -8
1/6 | -1/8 | -1 | -24)
Subtract 1/3 × (row 1) from row 2:
(1 | 1/2 | 1/3 | 18
0 | 7/12 | -7/9 | -14
1/6 | -1/8 | -1 | -24)
Multiply row 1 by 6:
(6 | 3 | 2 | 108
0 | 7/12 | -7/9 | -14
1/6 | -1/8 | -1 | -24)
Multiply row 2 by 36/7:
(6 | 3 | 2 | 108
0 | 3 | -4 | -72
1/6 | -1/8 | -1 | -24)
Multiply row 3 by 24:
(6 | 3 | 2 | 108
0 | 3 | -4 | -72
4 | -3 | -24 | -576)
Subtract 2/3 × (row 1) from row 3:
(6 | 3 | 2 | 108
0 | 3 | -4 | -72
0 | -5 | -76/3 | -648)
Multiply row 3 by -3:
(6 | 3 | 2 | 108
0 | 3 | -4 | -72
0 | 15 | 76 | 1944)
Swap row 2 with row 3:
(6 | 3 | 2 | 108
0 | 15 | 76 | 1944
0 | 3 | -4 | -72)
Subtract 1/5 × (row 2) from row 3:
(6 | 3 | 2 | 108
0 | 15 | 76 | 1944
0 | 0 | -96/5 | -2304/5)
Multiply row 3 by -5/96:
(6 | 3 | 2 | 108
0 | 15 | 76 | 1944
0 | 0 | 1 | 24)
Subtract 76 × (row 3) from row 2:
(6 | 3 | 2 | 108
0 | 15 | 0 | 120
0 | 0 | 1 | 24)
Divide row 2 by 15:
(6 | 3 | 2 | 108
0 | 1 | 0 | 8
0 | 0 | 1 | 24)
Subtract 3 × (row 2) from row 1:
(6 | 0 | 2 | 84
0 | 1 | 0 | 8
0 | 0 | 1 | 24)
Subtract 2 × (row 3) from row 1:
(6 | 0 | 0 | 36
0 | 1 | 0 | 8
0 | 0 | 1 | 24)
Divide row 1 by 6:
(1 | 0 | 0 | 6
0 | 1 | 0 | 8
0 | 0 | 1 | 24)
Collect results:
Answer: |x = 6 y = 8 z = 24
